ee 
ON THEORETICAL DYNAMICS. 23 
One integral is U=a, and if there be another integral V=é where V is a 
function of x,y, x',y' only, then 2',y' being by means of these two integrals 
expressed as a function of 2, y,a, 6, it is shown that xz'dx+y'dy is an exact 
differential, and putting J(aldet y'dy)=8, then that = f is a new integral 
of the given equations ; and in the case where d is a function of ¢ only, the 
= ae .. dO 
remaining integral is Fa deta. 
42. Binet’s memoir of 1841 contains an exposition of the theory of the 
variation of the arbitrary constants as applied to the general system of 
equations . 
d dF _ dF dL. dM 
ia get Marie cbs 
where F is any function of é, and of the coordinates a, y,z.. of the different 
points of the system, and of their differential coefficients 2’, y', 2', &c., and. 
L=0, M=0, &c. are any equations of equation between the coordinates 
2, Y,2,-. of the different points of the system, these equations may contain 
é, but they must not contain the differential coefficients z',y',z',.. The form 
is a more general one than that considered by Lagrange and Poisson. The 
memoir contains an elegant investigation of the variations of the elements of 
the orbit of a body acted upon by a central force, the expressions for the 
variations being obtained in a canonical form; and there is also a discussion 
of the problem suggested in Poisson’s report of 1830 on the manuscript work 
of Ostrogradsky. 
43. Jacobi’s note of 1842, in the ‘ Comptes Rendus,’ announces the general 
principle (being a particular case of the theorem of the ultimate multiplier) 
stated and demonstrated in the memoir next referred to, and gives also the 
rule for the formation of the multiplier in the case to which the general 
principle applies. 
44, Jacobi’s memoir of 1842, ‘De Motu Puncti singularis’: the author 
_ remarks, that the greater the difficulties in the general integration of the 
equations of dynamics, the greater the care which should be bestowed on 
the examination of the dynamical problems in which the integration can be 
reduced to quadratures; and the object of the memoir is stated to be the 
examination of the simplest case of all, viz. the problems relating to the 
motion of asingle point. The first section, entitled, ‘De Extensione quadam 
Principii Virium vivarum,’ contains a remark which, though obvious enough, 
is of considerable importance: the forces X, Y, Z which act upon a particle, 
may be such that Xdx+ Ydy-+ Zdz is not an exact differential, so that if the 
particle were free, there would be no force function, and the equations of 
motion would not be expressible in the standard form. But if the point 
move on a surface or a curve, then in the former case Xdx+ Ydy+ Zdz 
will be reducible to the form Pdp-+ Qdgq, which will be an exact differential 
if a single condition (instead of the three conditions which are required in 
the case of a free particle) be satisfied, and in the latter case it will be 
reducible to the form Pdp, which is, per se, an exact differential. In the 
case of a surface, the requisite transformation is given by the Hamiltonian 
form of the equations of motion, which Jacobi demonstrates for the case in 
hand; and then in the third section, with a view to its application to the 
particular case, he enumerates the general proposition “ que pro novo prin- 
cipio mechanico haberi potest,” which is as follows :— 
Consider the motion of a system of material points subjected to any 
